#2498462

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,15)(13,14)(18,20)(19,23)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,24,23,22)(26,27)(28,35)(32,34)(36,40);(10,13)(15,16)(17,19,21,23)(18,22,20,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,16,13,15)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,22,23,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=h,d^2=gh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2405658);